Gage R&R focuses on two key aspects of measurement:
Repeatability: Repeatability is the variation between successive measurements of the same part or trait by the same person using the same gage. In other words, how much variation do we see in measurements taken by the same person, on the same part, using the same tool?
Reproducibility: Reproducibility is the difference in the average of the measurements made by different people using the same instrument when measuring the identical characteristics on the same part. In other words, how much variation do we see in measurements taken by different people on the same part using the same tool?
(Ted Hessing, R.P. (2024) Gage repeatability and reproducibility (R&R))
In our analysis we conducted a Crossed Gage R&R
Crossed-gage R&R is when each operator measures each part, and it must have a balanced design with random factors. It is used for non-destructive testing.
In this Gage R&R study, the measurement system analysis was conducted using Vernier calipers, to evaluate repeatability and reproducibility among operators and equipment.
Product description
Flow chart Showing the Gage R&R process
# Data for 10 parts, 3 operators and 2 measurements per part
Operator<- factor(rep(1:3, each = 20))
Part<- factor(rep(rep(1:10, each = 2), 3))
# When inputting #operator data: part 1, part 1, part 2, part 2 etc.
Diameter<-c(49.28, 49.27, 49.36, 49.42, 49.26, 49.25, 46.35, 46.36, 49.26, 49.24,#op1
45.82, 45.81, 49.25, 49.27, 47.45, 47.45, 49.38, 49.37, 49.40, 49.44,
##########################
49.28, 49.35, 49.40, 49.37, 49.26, 49.26, 46.39, 46.41, 49.27, 49.29,#op2
45.75, 45.86, 49.30, 49.35, 47.47, 47.58, 49.39, 49.42, 49.44, 49.43,
##########################
49.28, 49.27, 49.37, 49.39, 49.34, 49.34, 46.44, 46.42, 49.32, 49.31,#op3
45.83, 45.91, 49.34, 49.28, 47.51, 47.54, 49.47, 49.36, 49.50, 49.45)
Data<-data.frame(Part,Operator,Diameter); print(Data)
#Load package
library("SixSigma")
#Perform gage R & R
ss.rr(var = Diameter, part = Part, appr = Operator, data = Data,
main = "Six Sigma Gage R&R Study",
sub = "",
alphaLim = 0.05,
errorTerm = "interaction",
digits = 4,
method = "crossed",
print_plot = TRUE,
signifstars = TRUE)
## Part Operator Diameter
## 1 1 1 49.28
## 2 1 1 49.27
## 3 2 1 49.36
## 4 2 1 49.42
## 5 3 1 49.26
## 6 3 1 49.25
## 7 4 1 46.35
## 8 4 1 46.36
## 9 5 1 49.26
## 10 5 1 49.24
## 11 6 1 45.82
## 12 6 1 45.81
## 13 7 1 49.25
## 14 7 1 49.27
## 15 8 1 47.45
## 16 8 1 47.45
## 17 9 1 49.38
## 18 9 1 49.37
## 19 10 1 49.40
## 20 10 1 49.44
## 21 1 2 49.28
## 22 1 2 49.35
## 23 2 2 49.40
## 24 2 2 49.37
## 25 3 2 49.26
## 26 3 2 49.26
## 27 4 2 46.39
## 28 4 2 46.41
## 29 5 2 49.27
## 30 5 2 49.29
## 31 6 2 45.75
## 32 6 2 45.86
## 33 7 2 49.30
## 34 7 2 49.35
## 35 8 2 47.47
## 36 8 2 47.58
## 37 9 2 49.39
## 38 9 2 49.42
## 39 10 2 49.44
## 40 10 2 49.43
## 41 1 3 49.28
## 42 1 3 49.27
## 43 2 3 49.37
## 44 2 3 49.39
## 45 3 3 49.34
## 46 3 3 49.34
## 47 4 3 46.44
## 48 4 3 46.42
## 49 5 3 49.32
## 50 5 3 49.31
## 51 6 3 45.83
## 52 6 3 45.91
## 53 7 3 49.34
## 54 7 3 49.28
## 55 8 3 47.51
## 56 8 3 47.54
## 57 9 3 49.47
## 58 9 3 49.36
## 59 10 3 49.50
## 60 10 3 49.45
## Complete model (with interaction):
##
## Df Sum Sq Mean Sq F value Pr(>F)
## Part 9 105.15 11.683 11076.17 < 2e-16 ***
## Operator 2 0.02 0.012 11.51 0.000603 ***
## Part:Operator 18 0.02 0.001 0.94 0.543185
## Repeatability 30 0.03 0.001
## Total 59 105.23
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## alpha for removing interaction: 0.05
##
##
## Reduced model (without interaction):
##
## Df Sum Sq Mean Sq F value Pr(>F)
## Part 9 105.15 11.683 10654.15 < 2e-16 ***
## Operator 2 0.02 0.012 11.07 0.000111 ***
## Repeatability 48 0.05 0.001
## Total 59 105.23
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Gage R&R
##
## VarComp %Contrib
## Total Gage R&R 0.0016487674 0.08
## Repeatability 0.0010965972 0.06
## Reproducibility 0.0005521701 0.03
## Operator 0.0005521701 0.03
## Part-To-Part 1.9470351659 99.92
## Total Variation 1.9486839333 100.00
##
## StdDev StudyVar %StudyVar
## Total Gage R&R 0.04060502 0.2436301 2.91
## Repeatability 0.03311491 0.1986895 2.37
## Reproducibility 0.02349830 0.1409898 1.68
## Operator 0.02349830 0.1409898 1.68
## Part-To-Part 1.39536202 8.3721721 99.96
## Total Variation 1.39595270 8.3757162 100.00
##
## Number of Distinct Categories = 48
Gage R&R Acceptance Criteria
(Interpreting Minitab’s Gage R&R chart - business performance improvement (BPI) (2023)
Total Gage R&R % Contribution: 0.08% (Below 1% → Acceptable)
Total Gage R&R % Study Variation: 2.91% (Below 10% → Acceptable)
Number of Distinct Categories (NDC): 48 (Greater than 10 → Acceptable)
Part-to-Part Variation: 99.92% (Ideal—most variation comes from actual part differences).
Operator Effect: Statistically significant (p = 0.000111) but practically small (0.03% contribution).
Part2Part Variation at 100% is high showing that the measurement system recognize the variation of each part across 3 different operators.
Gage R&R , Repeatability and Reproduciblty are all low indicating minimal error amongst the operator and the equipment
library(qcc)
# Data from the provided 10 results
cap_data <- c(49.28, 49.36, 49.26, 46.35, 49.26,
45.82, 49.25, 47.45, 49.38, 49.40)
# Create a qcc object for process capability analysis
xbar <- qcc(cap_data, type = "xbar.one", title = "Capability Study")
# Process Capability Analysis
process.capability(xbar, spec.limits = c(47, 51), target = 50)
Results
##
## Process Capability Analysis
##
## Call:
## process.capability(object = xbar, spec.limits = c(47, 51), target = 50)
##
## Number of obs = 10 Target = 50
## Center = 48.48 LSL = 47
## StdDev = 1.637 USL = 51
##
## Capability indices:
##
## Value 2.5% 97.5%
## Cp 0.4072 0.22306 0.5920
## Cp_l 0.3015 0.09243 0.5107
## Cp_u 0.5129 0.24907 0.7767
## Cp_k 0.3015 0.05237 0.5507
## Cpm 0.2985 0.14501 0.4532
##
## Exp<LSL 18% Obs<LSL 20%
## Exp>USL 6.2% Obs>USL 0%
Process Capability Summary
(Drroopesh and Drroopesh (2025)
Reflection)
DECLARATION: I declare that:
This work is entirely my own, and no part of it has been copied from any other person’s words or ideas, except as specifically acknowledged through the use of inverted commas and in-text references;
No part of this assignment has been written for me by any other person except where such collaboration has been authorised by the lecturer concerned;
I understand that I am bound by DkIT Academic Integrity Policy. I understand that I may be penalised if I have violated the policy in any way;
I have not used generative artificial intelligence (AI) (e.g. ChatGPT) unless it has been permitted by the lecturer(s) concerned;
This assignment has not been submitted for any other module at DkIT or any other institution, unless authorised by the relevant Lecturer(s);
I have read and abided by all of the requirements set down for this assignment.
TYPED SIGNATURE:Niall O’Callaghan
DATE: 03/03/25
Ted Hessing, R.P. (2024) Gage repeatability and reproducibility (R&R), Six Sigma Study Guide. Available at: https://sixsigmastudyguide.com/repeatability-and-reproducibility-rr/ (Accessed: 03 March 2025).
Interpreting Minitab’s Gage R&R chart - business performance improvement (BPI) (2023) Business Performance Improvement (BPI) - Helping businesses and organizations achieve Triple Bottom Line. Available at: https://www.biz-pi.com/interpreting-minitabs-gage-rr-chart/ (Accessed: 05 March 2025).
Drroopesh and Drroopesh (2025) Reflection: A basic guide, communitymedicine4all. Available at: https://communitymedicine4asses.wordpress.com/2025/01/10/reflection-a-basic-guide/ (Accessed: 06 March 2025).
BOSCH (2020). Capability of Measurement and Test Processes . [online] Available at: https://assets.bosch.com/media/global/bosch_group/purchasing_and_logistics/inform ation_for_business_partners/downloads/quality_docs/general_regulations/bosch_publi cations/booklet-no10-capability-of-measurement-and-test-processes_EN.pdf [Accessed 24 Feb. 2025].